On non-Kupka points of codimension one foliations on ℙ3

Abstract

We study the singular set of a codimension one holomorphic foliation on P 3 . We find a local normal form for these foliations near a codimension two component of the singular set that is not of Kupka type. We also determine the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on P 3 .

Codimension One Holomorphic Distributions on the Projective Three-space

“…In particular, either [12] the Baum-Bott residue BB(F , Z) along Z vanishes, providing a contradiction with the Baum-Bott formula 30, which yields It remains for us to show that K(F ) ∩ Z ⊆ Z is closed. Let p ∈ Z be a point in the closure K(F ) ∩ Z; in a small neighborhood V of p, we can take a disc Σ transversal to F , and we have a foliation…”

“…then by[12] the Baum-Bott residue BB(F , Z) along Z vanishes, providing a contradiction with the Baum-Bott formula 30, which yields…”

“…The tangent sheaf is locally free at 0. See [13] or [12,Theorem 4.1]. Finally, we recall a particular case of the Baum-Bott residues formula for a codimension one foliation F on a complex compact manifold X.…”

On the geometry of the singular locus of a codimension one foliation in $\mathbb P^n$

“…Proof. The proof in P 2 , may be found in an appendix of [CCF16] in the case c 1 (F ) = 0. We include the proof for completness.…”